Recall that a linear map \(\phi : \mathcal {A}\rightarrow \mathcal {B}(\mathcal {H})\)  is decomposable if there exists a Hilbert space \(\mathcal {K}\) , a bounded linear operator \(v: \mathcal {H}\rightarrow \mathcal {K}\) , and a Jordan \(*\) -homomorphism \(\Phi \) , i.e., \(\Phi (aa^{\prime }+a^{\prime }a) = \Phi (a)\Phi (a^{\prime })+\Phi (a^{\prime })\Phi (a)\)  and \(\Phi ^*(a) = \Phi (a^*)\)  for all \(a,a^{\prime } \in \mathcal {A}\)  (for details, see Sec. REF ), such that \(\phi = v^* \Phi v\) . Such maps are more general than quantum channels \(\phi : \mathcal {A}\rightarrow \mathcal {B}(\mathcal {H})\) , which are of similar form: \(\phi = v^*\Phi v\)  with \(\Phi \)  a \(C^*\) -homomorphism. By Stinespring's theorem [1]}, the latter is equivalent to \(\phi \)  being completely positive: if \(x_{ij} \in M_n(\mathcal {A})_+ = (M_n( \otimes \mathcal {A})_+\) , then \(\phi (x_{ij}) := \mathrm {id}_{M_n(} \otimes \phi (x_{ij}) \in M_n(\mathcal {B}(\mathcal {H}))_+\) . Similarly, decomposable maps can be characterised by a weaker positivity condition [2]}: if \(x_{ij} \in M_n(\mathcal {A})_+\)  and \(x_{ji} \in M_n(\mathcal {A})_+\) , then \(\phi (x_{ij}) \in M_n(\mathcal {B}(\mathcal {H}))_+\) . Let \(\mathcal {S}_D(\mathcal {A}\otimes \mathcal {B})\)  denote the class of bipartite states corresponding to decomposable maps under the Choi-Jamiołkowski isomorphism. Interestingly, (the weaker positivity condition in) \(\mathcal {S}_D(\mathcal {A}\otimes \mathcal {B})\)  is preserved under partial transposition (see also [3]}).
